Stokes%20phenomenon
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1: 36.5 Stokes Sets
§36.5 Stokes Sets
►§36.5(i) Definitions
… ►§36.5(ii) Cuspoids
… ►Elliptic Umbilic Stokes Set (Codimension three)
… ►§36.5(iv) Visualizations
…2: Bibliography P
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Stokes phenomenon demystified.
Bull. Inst. Math. Appl. 31 (1-2), pp. 21–28.
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Smoothing of the Stokes phenomenon for high-order differential equations.
Proc. Roy. Soc. London Ser. A 436, pp. 165–186.
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Smoothing of the Stokes phenomenon using Mellin-Barnes integrals.
J. Comput. Appl. Math. 41 (1-2), pp. 117–133.
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The Stokes phenomenon associated with the Hurwitz zeta function
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Proc. Roy. Soc. London Ser. A 461, pp. 297–304.
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Automatic computation of Bessel function integrals.
Comput. Phys. Comm. 25 (3), pp. 289–295.
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3: 2.11 Remainder Terms; Stokes Phenomenon
§2.11 Remainder Terms; Stokes Phenomenon
… ►§2.11(iv) Stokes Phenomenon
… ►That the change in their forms is discontinuous, even though the function being approximated is analytic, is an example of the Stokes phenomenon. Where should the change-over take place? Can it be accomplished smoothly? … ►For higher-order Stokes phenomena see Olde Daalhuis (2004b) and Howls et al. (2004). …4: 7.20 Mathematical Applications
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►For applications of the complementary error function in uniform asymptotic approximations of integrals—saddle point coalescing with a pole or saddle point coalescing with an endpoint—see Wong (1989, Chapter 7), Olver (1997b, Chapter 9), and van der Waerden (1951).
►The complementary error function also plays a ubiquitous role in constructing exponentially-improved asymptotic expansions and providing a smooth interpretation of the Stokes phenomenon; see §§2.11(iii) and 2.11(iv).
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5: 8.22 Mathematical Applications
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8.22.1
►plays a fundamental role in re-expansions of remainder terms in asymptotic expansions, including exponentially-improved expansions and a smooth interpretation of the Stokes phenomenon.
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6: Bibliography I
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The real roots of Bernoulli polynomials.
Ann. Univ. Turku. Ser. A I 37, pp. 1–20.
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Quasi-linear Stokes phenomenon for the second Painlevé transcendent.
Nonlinearity 16 (1), pp. 363–386.
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7: 10.46 Generalized and Incomplete Bessel Functions; Mittag-Leffler Function
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►For exponentially-improved asymptotic expansions in the same circumstances, together with smooth interpretations of the corresponding Stokes phenomenon (§§2.11(iii)–2.11(v)) see Wong and Zhao (1999b) when , and Wong and Zhao (1999a) when .
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►This reference includes exponentially-improved asymptotic expansions for when , together with a smooth interpretation of Stokes phenomena.
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8: Bibliography K
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Essential singularity of the Painlevé function of the second kind and the nonlinear Stokes phenomenon.
Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov.
(LOMI) 187, pp. 139–170 (Russian).
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Quasi-linear Stokes phenomenon for the Painlevé first equation.
J. Phys. A 37 (46), pp. 11149–11167.
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Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library.
ACM Trans. Math. Software 20 (4), pp. 447–459.
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Methods of computing the Riemann zeta-function and some generalizations of it.
USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.
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Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I.
Inverse Problems 20 (4), pp. 1165–1206.
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9: 6.16 Mathematical Applications
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§6.16(i) The Gibbs Phenomenon
… ►This nonuniformity of convergence is an illustration of the Gibbs phenomenon. … ► … ►10: Bibliography B
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Stokes surfaces of diffraction catastrophes with codimension three.
Nonlinearity 3 (2), pp. 281–291.
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Overlapping Stokes smoothings: Survival of the error function and canonical catastrophe integrals.
Proc. Roy. Soc. London Ser. A 444, pp. 201–216.
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Uniform asymptotic smoothing of Stokes’s discontinuities.
Proc. Roy. Soc. London Ser. A 422, pp. 7–21.
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Infinitely many Stokes smoothings in the gamma function.
Proc. Roy. Soc. London Ser. A 434, pp. 465–472.
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Stieltjes transforms and the Stokes phenomenon.
Proc. Roy. Soc. London Ser. A 429, pp. 227–246.
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